5L 2s (3/1-equivalent)

↖ 4L 1s⟨3/1⟩	↑ 5L 1s⟨3/1⟩	6L 1s⟨3/1⟩ ↗
← 4L 2s⟨3/1⟩	5L 2s (3/1-equivalent)	6L 2s⟨3/1⟩ →
↙ 4L 3s⟨3/1⟩	↓ 5L 3s⟨3/1⟩	6L 3s⟨3/1⟩ ↘

┌╥╥╥┬╥╥┬┐ │║║║│║║││ │││││││││ └┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LLLsLLs sLLsLLL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	4\7 to 3\5 (1086.8 ¢ to 1141.2 ¢)
Dark	2\5 to 3\7 (760.8 ¢ to 815.1 ¢)
Related MOS scales
Parent	2L 3s⟨3/1⟩
Sister	2L 5s⟨3/1⟩
Daughters	7L 5s⟨3/1⟩, 5L 7s⟨3/1⟩
Neutralized	3L 4s⟨3/1⟩
2-Flought	12L 2s⟨3/1⟩, 5L 9s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	4\7 (1086.8 ¢)
Supersoft (L:s = 4:3)	15\26 (1097.3 ¢)
Soft (L:s = 3:2)	11\19 (1101.1 ¢)
Semisoft (L:s = 5:3)	18\31 (1104.4 ¢)
Basic (L:s = 2:1)	7\12 (1109.5 ¢)
Semihard (L:s = 5:2)	17\29 (1114.9 ¢)
Hard (L:s = 3:1)	10\17 (1118.8 ¢)
Superhard (L:s = 4:1)	13\22 (1123.9 ¢)
Collapsed (L:s = 1:0)	3\5 (1141.2 ¢)

5L 2s⟨3/1⟩, also called triatonic, is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 5 large steps and 2 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1086.8 ¢ to 1141.2 ¢, or from 760.8 ¢ to 815.1 ¢.

Name

The name triatonic was coined by CompactStar, and is a back-formation from "diatonic" with di- being interpreted as 2 (the octave) and replaced with tri- for 3 (the tritave). It is not an official name in TAMNAMS.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

MOS data is deprecated. Please use the following templates individually: MOS intervals, MOS genchain, and MOS mode degrees

Theory

As a macrodiatonic scale

5L 2s⟨3/1⟩ is a macrodiatonic scale with the period of a tritave. This means it is a diatonic scale, but its intervals are all stretched to the point of being unrecognizable—the diatonic fifth is now the size of a major seventh, and octaves are stretched out to tritaves. Interestingly, 19edt, an approximation of 12edo, has a tuning of this scale, meaning it contains both a diatonic scale (which approximates 12edo's diatonic scale) and a triatonic scale.

Temperament interpretations

It is possible to construct no-twos rank-2 temperament interpretations of this scale, although most of these do not fit neatly into the 3.5.7 subgroup used for Bohlen–Pierce. Two intervals that can serve as macrodiatonic generators are ~17/9, which is just near 19edt in the soft range, and ~21/11 which is just near 17edt in the hard range.

Very soft scales (in the range between 26edt and 45edt, serving as a macro-flattone) can be interpreted in the 3.5.7.17 subgroup as Mizar, in which the generator of a flattened ~17/9 stacks twice and tritave-reduces to 25/21, which generates Sirius temperament. Scales close to basic have an interpretation in the 3.13.17 subgroup, documented as Sadalmelik in which the generator (the stretched counterpart of the fifth) is also ~17/9 and a stack of 4 generators tritave-reduced (equivalent to the major third) is ~13/9; see also the page for 12edt. Harder scales can be interpreted in Mintaka temperament in the 3.7.11 subgroup, which tempers out 1331/1323 so that the dark generator (the stretched counterpart of the fourth) is ~11/7, a stack of 2 generators (equivalent to the minor seventh) is ~27/11, and a stack of three generators (equivalent to the minor third) is ~9/7.

Notation

Being a macrodiatonic scale, it can notated using the traditional diatonic notation, if all intervals are reinterpreted as their stretched versions (like octaves as tritaves). However, this approach involves 1-based indexing for a non-diatonic MOS which is generally discouraged. Alternatively, a generic MOS notation may be used like diamond-mos notation, which enables 0-based indexing at the cost of obscuring the connection to the standard diatonic scale.

Intervals

Intervals of 5L 2s⟨3/1⟩

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Minor 1-mosstep	m1ms	s	0.0 ¢ to 271.7 ¢
	Major 1-mosstep	M1ms	L	271.7 ¢ to 380.4 ¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	380.4 ¢ to 543.4 ¢
	Major 2-mosstep	M2ms	2L	543.4 ¢ to 760.8 ¢
3-mosstep	Perfect 3-mosstep	P3ms	2L + s	760.8 ¢ to 815.1 ¢
	Augmented 3-mosstep	A3ms	3L	815.1 ¢ to 1141.2 ¢
4-mosstep	Diminished 4-mosstep	d4ms	2L + 2s	760.8 ¢ to 1086.8 ¢
	Perfect 4-mosstep	P4ms	3L + s	1086.8 ¢ to 1141.2 ¢
5-mosstep	Minor 5-mosstep	m5ms	3L + 2s	1141.2 ¢ to 1358.5 ¢
	Major 5-mosstep	M5ms	4L + s	1358.5 ¢ to 1521.6 ¢
6-mosstep	Minor 6-mosstep	m6ms	4L + 2s	1521.6 ¢ to 1630.2 ¢
	Major 6-mosstep	M6ms	5L + s	1630.2 ¢ to 1902.0 ¢
7-mosstep	Perfect 7-mosstep	P7ms	5L + 2s	1902.0 ¢